
 [1]}
\(A\subseteq 2^{\omega }\)  is universally Baire if and only if for each forcing notion \(P\)  there are
trees \(T_A,S_A\)  on \(2\times \delta \)  for some \(\delta > |P|\) 
such that \(A=p[[T_A]]\)  (where \(p:(2\times \kappa )^\omega \rightarrow 2^{\omega }\)  denotes the projection on the first component and \([T]\)  denotes the body of the tree \(T\) ), and
\(P\Vdash T_A\text{ and }S_A\text{ project to complements},\) 
by this meaning that for all \(G\)  \(V\) -generic for \(P\)
\(V[G]\models (p[[T_A]]\cap p[[S_A]]=\emptyset )\wedge (p[[T_A]]\cup p[[S_A]]=(2^\omega )^{V[G]})\) 

Any two Polish spaces \(X,Y\)  of the same cardinality are Borel isomorphic [2]}.

Any Polish space is Borel isomorphic to a Borel subset of \([0;1]^\omega \)  [2]}, hence also to a Borel subset of \(2^\omega \)  (by the previous item).

Given \(\phi :\mathbb {N}\rightarrow \mathbb {N}\) , \(\prod _{n\in \omega }2^{\phi (n)}\)  is compact and Polish
(it is actually homemomorphic to the union of \(2^\omega \)  with a countable Hausdorff space)
[2]}.

